The term "continuum" is perfectly valid. Aside from the "continuum espace-temps", which designates the four-dimensional space in which physics are set, there is the much less scientific expression "sur un continuum allant de ... à ...", which in English would translate as "on a spectrum from ... to ...".

I have never heard the plural form continua used.

As for the sentence that you were trying to translate originally, the correct translation is indeed mécanique des milieux continus. That's an expression that maps directly to the concept of continuum mechanics in English and is used e.g. to describe a course in a curriculum.

Similarly Special relativity does not translate to *"relativité spéciale" but to relativité restreinte ("restricted").